Stability of the Origin in 2x2 Systems of Differential Equations

Overview

In a system of differential equations defined by \mathbf{x}' = A \mathbf{x}, understanding the stability of the origin (the point \mathbf{x} = 0) is essential. The stability depends on the eigenvalues of the 2 \times 2 matrix A. Different configurations of eigenvalues lead to distinct types of stability, affecting how trajectories in the phase plane behave near the origin.

Types of Stability

Repelling Node (Unstable)

• Eigenvalues: 0 < \lambda_1 \leq \lambda_2 (both positive real numbers).
• Description: The origin is unstable. Trajectories move away from the origin in all directions. This configuration is known as a repelling node.

# Define the system of ODEs for a repelling node
def dx_dt(X, t):
    x1, x2 = X
    dx1 = 3 * x1 + 1 * x2
    dx2 = 1 * x1 + 3 * x2
    return [dx1, dx2]

# Generate grid for direction field
x1_vals = np.linspace(-5, 5, 35)
x2_vals = np.linspace(-5, 5, 35)
X1, X2 = np.meshgrid(x1_vals, x2_vals)

# Calculate slopes for direction field
U = 3 * X1 + 1 * X2
V = 1 * X1 + 3 * X2

# Normalize the direction field arrows
speed = np.sqrt(U**2 + V**2)
epsilon = 1e-10  # Small value to prevent division by zero
U_norm = U / (speed + epsilon)
V_norm = V / (speed + epsilon)

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Straight-line solutions (eigenvectors)
x = np.linspace(-5, 5, 200)
plt.plot(x, x, 'r', linewidth=3, label=r'Unstable direction, $\lambda=3$')
plt.plot(x, -x, 'b', linewidth=3, label=r'Unstable direction, $\lambda=4$')


# Set plot limits and labels
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Repelling Node (Unstable)", fontsize=18)

# Add grid and legend
plt.grid(True, linestyle='--', alpha=0.5)
plt.legend(loc="best", fontsize=12, framealpha=1.0, frameon=True)

# Display the plot
plt.tight_layout()
plt.show()

Saddle (Unstable)

• Eigenvalues: \lambda_1 < 0 < \lambda_2 (one positive and one negative real number).
• Description: The origin is unstable, with trajectories approaching along one direction and moving away along another. This configuration creates a “saddle” shape in the phase plane, and is called a saddle point.

# Define the system of ODEs for a saddle point
def dx_dt(X, t):
    x1, x2 = X
    dx1 = 1 * x1 + 2 * x2
    dx2 = 2 * x1 - 1 * x2
    return [dx1, dx2]

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Straight-line solutions
plt.plot(x, x, 'r', linewidth=3, label=r'Unstable direction, $\lambda=1$')
plt.plot(x, -x, 'b', linewidth=3, label=r'Stable direction, $\lambda=-1$')


# Labels and title
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Saddle Point (Unstable)", fontsize=18)

plt.grid(True, linestyle='--', alpha=0.5)
plt.legend(loc="best", fontsize=12, framealpha=1.0, frameon=True)
plt.tight_layout()
plt.show()

Attracting Node (Stable)

• Eigenvalues: \lambda_1 \leq \lambda_2 < 0 (both negative real numbers).
• Description: The origin is stable, and all trajectories are attracted toward it. This setup is called an attracting node, as trajectories converge to the origin from all directions.

# Define the system of ODEs
def dx_dt(X, t):
    x1, x2 = X
    dx1 = -3 * x1 + 2 * x2
    dx2 = 2 * x1 - 3 * x2
    return [dx1, dx2]

# Generate grid for direction field
x1_vals = np.linspace(-5, 5, 35)
x2_vals = np.linspace(-5, 5, 35)
X1, X2 = np.meshgrid(x1_vals, x2_vals)

# Calculate slopes for direction field
U = -3 * X1 + 2 * X2
V = 2 * X1 - 3 * X2

# Normalize the direction field arrows
speed = np.sqrt(U**2 + V**2)
epsilon = 1e-10  # Small value to prevent division by zero
U_norm = U / (speed + epsilon)
V_norm = V / (speed + epsilon)

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Plot the straight-line solutions (eigenvectors)
x = np.linspace(-5, 5, 200)
plt.plot(x, x, 'r', linewidth=3, label=r'Stable direction, $\lambda=-1$')
plt.plot(x, -x, 'b', linewidth=3, label=r'Stable direction, $\lambda=-5$')


# Set plot limits and labels
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Attracting Node (Stable)", fontsize=18)

# Add grid and legend
plt.grid(True, linestyle='--', alpha=0.5)
plt.legend(loc="best", fontsize=12, framealpha=1.0, frameon=True)

# Display the plot
plt.tight_layout()
plt.show()

Spiral Source (Unstable)

• Eigenvalues: \lambda = a \pm bi with a > 0 (complex eigenvalues with a positive real part).
• Description: The origin is unstable. Trajectories spiral outward away from the origin, creating a spiral source.

# Define the system of ODEs for a spiral source (unstable)
def dx_dt(X, t):
    x1, x2 = X
    dx1 = 1 * x1 - 2 * x2
    dx2 = 2 * x1 + 1 * x2
    return [dx1, dx2]

# Generate grid for direction field
x1_vals = np.linspace(-5, 5, 35)
x2_vals = np.linspace(-5, 5, 35)
X1, X2 = np.meshgrid(x1_vals, x2_vals)

# Calculate slopes for direction field
U = 1 * X1 - 2 * X2
V = 2 * X1 + 1 * X2

# Normalize the direction field arrows
speed = np.sqrt(U**2 + V**2)
epsilon = 1e-10  # Small value to prevent division by zero
U_norm = U / (speed + epsilon)
V_norm = V / (speed + epsilon)

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Additional trajectories to show the outward spiral behavior
initial_conditions = [
    [0, .1],
    [0, -.1],
    [.1, 0],
    [-.1, 0],
]

t_values = np.linspace(0, 5, 500)
for ic in initial_conditions:
    sol = odeint(dx_dt, ic, t_values)
    plt.plot(sol[:, 0], sol[:, 1], 'green', linestyle='-', linewidth=2)

# Set plot limits and labels
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Spiral Source (Unstable)", fontsize=18)

# Add grid
plt.grid(True, linestyle='--', alpha=0.5)

# Display the plot
plt.tight_layout()
plt.show()

Center (Neutral Stability)

• Eigenvalues: \lambda = a \pm bi with a = 0 (purely imaginary eigenvalues).
• Description: The origin neither attracts nor repels trajectories. Instead, they form closed orbits around the origin, resulting in a behavior called a center. This indicates neutral stability.

# Define the system of ODEs for a center (neutral stability)
def dx_dt(X, t):
    x1, x2 = X
    dx1 = -1 * x2
    dx2 = 1 * x1
    return [dx1, dx2]

# Generate grid for direction field
x1_vals = np.linspace(-5, 5, 35)
x2_vals = np.linspace(-5, 5, 35)
X1, X2 = np.meshgrid(x1_vals, x2_vals)

# Calculate slopes for direction field
U = -1 * X2
V = 1 * X1

# Normalize the direction field arrows
speed = np.sqrt(U**2 + V**2)
epsilon = 1e-10  # Small value to prevent division by zero
U_norm = U / (speed + epsilon)
V_norm = V / (speed + epsilon)

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Additional trajectories to show the circular motion
initial_conditions = [
    [5, 0],
    [0, 3],
    [1, 0]

]

t_values = np.linspace(0, 20, 500)
for ic in initial_conditions:
    sol = odeint(dx_dt, ic, t_values)
    plt.plot(sol[:, 0], sol[:, 1], 'green', linestyle='-', linewidth=2)

# Set plot limits and labels
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Center (Neutral Stability)", fontsize=18)

# Add grid
plt.grid(True, linestyle='--', alpha=0.5)

# Display the plot
plt.tight_layout()
plt.show()

Spiral Sink (Stable)

• Eigenvalues: \lambda = a \pm bi with a < 0 (complex eigenvalues with a negative real part).
• Description: The origin is stable, and trajectories spiral inward toward it, forming a spiral sink.

# Define the system of ODEs for a spiral sink (stable)
def dx_dt(X, t):
    x1, x2 = X
    dx1 = -1 * x1 - 2 * x2
    dx2 = 2 * x1 - 1 * x2
    return [dx1, dx2]

# Generate grid for direction field
x1_vals = np.linspace(-5, 5, 35)
x2_vals = np.linspace(-5, 5, 35)
X1, X2 = np.meshgrid(x1_vals, x2_vals)

# Calculate slopes for direction field
U = -1 * X1 - 2 * X2
V = 2 * X1 - 1 * X2

# Normalize the direction field arrows
speed = np.sqrt(U**2 + V**2)
epsilon = 1e-10  # Small value to prevent division by zero
U_norm = U / (speed + epsilon)
V_norm = V / (speed + epsilon)

# Plot the direction field
plt.figure(figsize=(8, 8))
plt.quiver(X1, X2, U_norm, V_norm, angles="xy", color="black", scale=40, alpha=0.4, width=0.0035)

# Additional trajectories to show the spiral inward behavior
initial_conditions = [
    [5, 0],
    [0, 5],
    [-5, 0],
    [0, -5],
]

t_values = np.linspace(0, 10, 500)
for ic in initial_conditions:
    sol = odeint(dx_dt, ic, t_values)
    plt.plot(sol[:, 0], sol[:, 1], 'green', linestyle='-', linewidth=2)

# Set plot limits and labels
plt.xlim(-5, 5)
plt.ylim(-5, 5)
plt.xlabel("$x_1$", fontsize=14)
plt.ylabel("$x_2$", fontsize=14)
plt.title("Spiral Sink (Stable)", fontsize=18)

# Add grid and legend
plt.grid(True, linestyle='--', alpha=0.5)


# Display the plot
plt.tight_layout()
plt.show()

Summary

The stability of the origin in a 2 \times 2 system of differential equations depends on the real and imaginary parts of the eigenvalues of matrix A:

• Real positive eigenvalues lead to a repelling node (unstable).
• Mixed positive and negative real eigenvalues create a saddle point (unstable).
• Real negative eigenvalues result in an attracting node (stable).
• Complex eigenvalues with a positive real part indicate a spiral source (unstable).
• Purely imaginary eigenvalues create a center (neutral stability).
• Complex eigenvalues with a negative real part lead to a spiral sink (stable).

By determining the eigenvalues, we can classify the type of stability at the origin and predict the behavior of trajectories in the phase plane.